Group #19156414 — subgroups

Subgroups — C₂₃×#8788 #19156414

Order 5888.

322 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1831	C₂	1	2	yes	maximal
2	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1277	C₂	1	4	yes	maximal
4	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
6	C₄³×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#336	—	2	4	no	—
8	C₂₃×#336	C₄	1	4	yes	—
9	C₂²×C₈×C₂₃	—	2	4	no	—
10	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
11	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
12	C₂×C₄²×C₂₃	D₄	1	2	yes	—
13	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
15	C₂₃×C₄⋊C₈	—	2	8	no	—
16	C₄×C₈×C₂₃	—	2	4	no	—
17	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
18	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₈×C₂₃	—	2	4	no	—
22	C₂×C₈×C₂₃	—	2	8	no	—
23	C₂×C₈×C₂₃	—	4	4	no	—
24	C₂×C₈×C₂₃	—	4	8	no	—
25	C₄²×C₂₃	—	2	4	no	—
26	C₄²×C₂₃	—	2	4	no	—
27	C₄²×C₂₃	—	4	2	no	—
28	C₄²×C₂₃	#41	1	4	yes	—
29	#8788	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	—	2	4	no	—
33	C₂×C₄×C₂₃	—	2	4	no	—
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	#117	1	2	yes	—
36	C₂×C₄×C₂₃	#124	1	2	yes	—
37	C₂×C₄×C₂₃	#125	1	2	yes	—
38	C₂×C₄×C₂₃	#131	1	4	yes	—
39	C₂×C₄×C₂₃	#135	1	4	yes	—
40	C₈×C₂₃	—	4	4	no	—
41	C₈×C₂₃	—	8	4	no	—
42	#1831	C₂×C₂₃	1	2	yes	—
43	#1744	C₂×C₂₃	1	1	yes	characteristic
44	#1277	C₂×C₂₃	1	4	yes	—
45	C₂²×C₂₃	#382	1	1	yes	characteristic
46	C₂²×C₂₃	#424	1	2	yes	—
47	C₂²×C₂₃	#427	1	2	yes	—
48	C₂²×C₂₃	#430	1	2	yes	—
49	C₄×C₂₃	—	2	2	no	—
50	C₄×C₂₃	—	4	4	no	—
51	C₄×C₂₃	#433	1	4	yes	—
52	C₄×C₂₃	#488	1	4	yes	—
53	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
54	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
55	C₄³	C₂²×C₂₃	1	1	yes	characteristic
56	#336	—	2	4	no	—
57	#336	C₄×C₂₃	1	4	yes	—
58	C₂×C₂₃	#1572	1	4	yes	—
59	C₂×C₂₃	#1832	1	1	yes	characteristic
60	C₂×C₂₃	#1835	1	1	yes	characteristic
61	C₂×C₂₃	#1838	1	1	yes	characteristic
62	C₂²×C₈	—	2	4	no	—
63	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
64	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
65	C₂×C₄²	D₄×C₂₃	1	2	yes	—
66	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
67	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
68	C₄⋊C₈	—	2	8	no	—
69	C₄×C₈	—	2	4	no	—
70	C₂₃	#8788	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
72	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
74	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	—	4	4	no	—
78	C₂×C₈	—	4	8	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	#41×C₂₃	1	4	yes	—
83	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	C₂₃×#117	1	2	yes	—
89	C₂×C₄	C₂₃×#124	1	2	yes	—
90	C₂×C₄	C₂₃×#125	1	2	yes	—
91	C₂×C₄	C₂₃×#131	1	4	yes	—
92	C₂×C₄	C₂₃×#135	1	4	yes	—
93	C₈	—	4	4	no	—
94	C₈	—	8	4	no	—
95	C₂²	C₂₃×#382	1	1	yes	characteristic
96	C₂²	C₂₃×#424	1	2	yes	—
97	C₂²	C₂₃×#427	1	2	yes	—
98	C₂²	C₂₃×#430	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	4	4	no	—
101	C₄	C₂₃×#433	1	4	yes	—
102	C₄	C₂₃×#488	1	4	yes	—
103	C₂	C₂₃×#1572	1	4	yes	—
104	C₂	C₂₃×#1832	1	1	yes	characteristic
105	C₂	C₂₃×#1835	1	1	yes	characteristic
106	C₂	C₂₃×#1838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1831

C₂

yes

maximal

C₂₃×#1744

C₂

yes

maximal, characteristic

C₂₃×#1277

C₂

yes

maximal

C₂×C₂₃×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₄³×C₂₃

C₂²

yes

characteristic

C₂₃×#336

—

C₂₃×#336

C₄

yes

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

Q₈

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

—

C₂²×C₄×C₂₃

C₄⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

#41

yes

—

#8788

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

#124

yes

—

C₂×C₄×C₂₃

#125

yes

—

C₂×C₄×C₂₃

#131

yes

—

C₂×C₄×C₂₃

#135

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

#1831

C₂×C₂₃

yes

—

#1744

C₂×C₂₃

yes

characteristic

#1277

C₂×C₂₃

yes

—

C₂²×C₂₃

#382

yes

characteristic

C₂²×C₂₃

#424

yes

—

C₂²×C₂₃

#427

yes

—

C₂²×C₂₃

#430

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#433

yes

—

C₄×C₂₃

#488

yes

—

C₂×C₄⋊C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₄³

C₂²×C₂₃

yes

characteristic

#336

—

#336

C₄×C₂₃

yes

—

C₂×C₂₃

#1572

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

—

C₂×C₄²

D₄×C₂₃

yes

—

C₂×C₄²

Q₈×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₂₃

#8788

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄⋊C₄×C₂₃

yes

—

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

C₂²×C₄

C₂×Q₈×C₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#41×C₂₃

yes

—

C₂³

C₂×C₄⋊C₄×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂₃×#117

yes

—

C₂×C₄

C₂₃×#124

yes

—

C₂×C₄

C₂₃×#125

yes

—

C₂×C₄

C₂₃×#131

yes

—

C₂×C₄

C₂₃×#135

yes

—

C₈

—

C₈

—

C₂²

C₂₃×#382

yes

characteristic

C₂²

C₂₃×#424

yes

—

C₂²

C₂₃×#427

yes

—

C₂²

C₂₃×#430

yes

—

C₄

—

100

C₄

—

101

C₄

C₂₃×#433

yes

—

102

C₄

C₂₃×#488

yes

—

103

C₂

C₂₃×#1572

yes

—

104

C₂

C₂₃×#1832

yes

characteristic

105

C₂

C₂₃×#1835

yes

characteristic

106

C₂

C₂₃×#1838

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

subgroup

G/H

center

C₂²×C₄×C₂₃

C₂×D₄

commutator

C₂²×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

socle, prime radical

C₂³×C₂₃

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

N	H ≅ G/N	type
29. #8788	C₂₃	direct (N × H)
70. C₂₃	#8788	direct (N × H)

H ≅ G/N

type

29. #8788

C₂₃

direct (N × H)

70. C₂₃

#8788

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#1831	2
C₂₃×#1744	2
C₂₃×#1277	2
#8788	23

subgroup

index

C₂₃×#1831

C₂₃×#1744

C₂₃×#1277

#8788

Subgroups — C₂₃×#8788 #19156414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8788 #19156414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8788 #19156414